Array of dipoles near a hyperbolic metamaterial: Evanescent-to-propagating Floquet wave transformation

We investigate the capabilities of hyperbolic metamaterials (HMs) to couple near-fields (i.e., evanescent waves) emitted by a two-dimensional periodic array of electric dipoles to propagating waves. In particular, large-order Floquet harmonics with transverse-magnetic polarization that would be evanescent in free space, and therefore confined near the array surface, are transformed into a propagating spectrum inside the HM and thus carry power away. Because of this property, independent of the finite or infinite extent of the HM, the power generated by an array of elementary electric dipoles is strongly enhanced and is mostly directed into the HM when the array is located near a HM surface. In particular, the power coupled to the HM exhibits narrow frequency features that can be employed in detection applications. The results shown in this paper provide a clear signature on wave dynamics in HMs. A link between the results pertaining to the case of an isolated dipole on top of HM and the planar array is found to be convenient in explaining both wave dynamics and spectral power distribution. The narrow frequency emission features appear in the array case only; they depend on its spatial periodicity and remarkably on the HM thickness.

Figure 1

(a) Schematic of an array of electric dipoles at a distance $h$ from the surface of a hyperbolic metamaterial. Example of HM made of a stack of dielectric and silver layers with thicknesses $d_1$ and $d_2$ and relative permittivities ${\varepsilon }_1$ and ${\varepsilon }_2$. (b) Schematic of single dipole at a distance $h$ from the surface of a hyperbolic metamaterial as in part (a).

Figure 2

(a) Enhancement of the power $P=P_{\mathrm{up}}+P_{\mathrm{down}}$ emitted by an array of dipoles ($a=b=300\mathrm{nm}$) with a HM underneath, with respect to that emitted in free space. Dipoles are polarized along $x$, with ${\mathbf{k}}_{t,00}=0.5k_0\widehat{\mathbf{x}}$. For comparison, the power enhancement pertaining to a single dipole over the HM is also provided. (b) Ratio of power emitted by the array and a single dipole toward HM (down) and toward the upper homogeneous isotropic space (up) versus frequency. Dashed lines are obtained for a homogeneous HM (via EMA) made of silver and silica layers with equal thicknesses $d_1=d_2=10\mathrm{nm}$, whereas solid lines are obtained using a rigorous multilayer Green's function implementation for the same HM. Silver permittivity ${\varepsilon }_2$ is from [50] and dielectric relative permittivity ${\varepsilon }_1$ is equal to 2.2. Source distance from the HM is assumed as $h=10\mathrm{nm}$.

Figure 3

(a) Emitted power $P=P_{\mathrm{up}}+P_{\mathrm{down}}$ by an array of dipoles (polarized along $x$ and with ${\mathbf{k}}_{t,00}=0.5k_0\widehat{\mathbf{x}}$) and (b) the ratio of power emitted by the array toward HM and toward free space versus frequency with respect to different array periods $a=b$. The HM is as in Fig. 2. This result is calculated assuming a multilayer HM.

Figure 4

Total spectral power versus $k_x$ and $k_y$. (a, b) ${log}_{10}\left[\left(U_{\mathrm{up}}^{\mathrm{TM}}+U_{\mathrm{down}}^{\mathrm{TM}}+U_{\mathrm{up}}^{\mathrm{TE}}+U_{\mathrm{down}}^{\mathrm{TE}}\right)/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$ emitted by the unit transverse electric dipole ${\mathbf{p}}_{00}=\left(\widehat{\mathbf{x}}+\widehat{\mathbf{y}}\right)//\sqrt{2}$ Cm; (c, d) ${log}_{10}\left[\left(W_{\mathrm{up}}^{\mathrm{TM}}+W_{\mathrm{down}}^{\mathrm{TM}}\right)/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$ emitted by the unit vertical dipole ${\mathbf{p}}_{00}=\widehat{\mathbf{z}}$. In (a) and (c), EMA is used in HM modeling, whereas in (b) and (d), multilayer HM is assumed at 650 THz.

Figure 5

Spectral power ${log}_{10}\left[U_{\mathrm{up}/\mathrm{down}}^{\mathrm{TM}/\mathrm{TE}}/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$ versus $k_x$ and $k_y$ emitted by the unit transverse dipole ${\mathbf{p}}_{00}=\left(\widehat{\mathbf{x}}+\widehat{\mathbf{y}}\right)//\sqrt{2}$ Cm in (a) $U_{\mathrm{up}}^{\mathrm{TM}}$: TM polarization and +$z$ direction, (b) $U_{\mathrm{down}}^{\mathrm{TM}}$: TM polarization and −$z$ direction, (c) $U_{\mathrm{up}}^{\mathrm{TE}}$: TE polarization and +$z$ direction, and (d) $U_{\mathrm{down}}^{\mathrm{TE}}$: TE polarization and −$z$ direction. This result is calculated assuming a multilayer HM.

Figure 6

(a, c, e) The spectral power of the Floquet harmonics versus the indices $p$ and $q$ for the array periods $a=b=$ 150, 300, 600 nm; (b, d, f) total spectral power ${log}_{10}\left[\left(U_{\mathrm{up}}^{\mathrm{TM}}+U_{\mathrm{down}}^{\mathrm{TM}}+U_{\mathrm{up}}^{\mathrm{TE}}+U_{\mathrm{down}}^{\mathrm{TE}}\right)/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$ versus $k_x$ and $k_y$ emitted by the unit transverse dipole ${\mathbf{p}}_{00}=\left(\widehat{\mathbf{x}}+\widehat{\mathbf{y}}\right)/\sqrt{2}$ Cm at 650 THz, where the white circles denote the Floquet harmonic sampling locations on the $k_x$–$k_y$ plane in the array case for various array periods, as in (a, c, e). This result is calculated assuming a multilayer HM.

Figure 7

HM substrate with finite thickness, where $N$ is the number of metal–dielectric bilayers. This substrate configuration is investigated for both a single dipolar source and an array of sources.

Figure 8

Spectral power ${log}_{10}\left[U_{\mathrm{down}}^{\mathrm{TM}}/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$ versus $k_x$ and $k_y$ emitted by the unit transverse dipole ${\mathbf{p}}_{00}=\widehat{\mathbf{x}}$ Cm over a multilayer HM at 650 THz for a varying number of bilayers $N$.

Figure 9

Emitted power by an array of dipoles (normalized to the power emitted by the same array in free space) and ratio of power emitted toward HM and toward free space versus frequency, compared with the case of a single dipole on HM. This result is calculated assuming a multilayer HM.

Figure 10

Power spectrum of extraordinary TM waves carrying power in the −$z$ direction, ${log}_{10}\left[U_{\mathrm{down}}^{\mathrm{TM}}\left({\mathbf{k}}_t\right)/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$, versus $k_x$ (horizontal axis) and frequency (vertical axis) emitted by a single transverse dipole with ${\mathbf{p}}_{00}=\widehat{\mathbf{x}}$ Cm on top of a finite thickness of HM with $N$ = 5 layers. Black dashed curves indicate spectral sampling lines corresponding to ${\mathbf{k}}_{t,p0}=k_{x,p}\widehat{\mathbf{x}}$ for $p=-8,-7,...,0,...,7,8$ (denoted at top of the plot) when an array of dipoles in considered. This result is calculated assuming a multilayer HM.

Figure 11

The schematic of (a) a periodic array of rectangular current sheets on top of HM and (b) a single rectangular current sheet on top of HM.

Figure 12

Spectral power ${log}_{10}\left[\left(U_{\mathrm{down}}^{\mathrm{TM}}\right)/\left({\mathrm{Wm}}^2{\mathrm{s}}^2\right)\right]$ versus $k_x$ and $k_y$ emitted at 650 THz (a) by the discrete dipole ${\mathbf{p}}_{00}=\left(\widehat{\mathbf{x}}+\widehat{\mathbf{y}}\right)/\sqrt{2}$ Cm and (b, c, d) by the constant sheet current ${\mathbf{J}}_{00}=-i\omega {\mathbf{p}}_{00}/\left(l_x{l}_y\right)$ flowing over a flat square sheet with sides $l_x=l_y=20,30,40\mathrm{nm}$, respectively, centered at the origin. This result is calculated assuming a multilayer HM.